Calculus Examples

$\frac{d x}{d y} + \frac{2 x y}{y^{2}} - \frac{\sin (y)}{y^{2}} = 0$

Step 1

Rewrite the differential equation as $\frac{d x}{d y} + M (y) x = Q (y)$ .

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Step 1.1

Add $\frac{\sin (y)}{y^{2}}$ to both sides of the equation.

$\frac{d x}{d y} + \frac{2 x y}{y^{2}} = \frac{\sin (y)}{y^{2}}$

Step 1.2

Factor $x$ out of $\frac{2 x y}{y^{2}}$ .

$\frac{d x}{d y} + x \frac{2 y}{y^{2}} = \frac{\sin (y)}{y^{2}}$

Step 1.3

Reorder $x$ and $\frac{2 y}{y^{2}}$ .

$\frac{d x}{d y} + \frac{2 y}{y^{2}} x = \frac{\sin (y)}{y^{2}}$

Step 2

The integrating factor is defined by the formula $e^{\int P (y) d y}$ , where $P (y) = \frac{2 y}{y^{2}}$ .

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Step 2.1

Set up the integration.

$e^{\int \frac{2 y}{y^{2}} d y}$

Step 2.2

Integrate $\frac{2 y}{y^{2}}$ .

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Step 2.2.1

Cancel the common factor of $y$ and $y^{2}$ .

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Step 2.2.1.1

Factor $y$ out of $2 y$ .

$e^{\int \frac{y \cdot 2}{y^{2}} d y}$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $y$ out of $y^{2}$ .

$e^{\int \frac{y \cdot 2}{y \cdot y} d y}$

Step 2.2.1.2.2

Cancel the common factor.

$e^{\int \frac{y \cdot 2}{y \cdot y} d y}$

Step 2.2.1.2.3

Rewrite the expression.

$e^{\int \frac{2}{y} d y}$

Step 2.2.2

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

$e^{2 \int \frac{1}{y} d y}$

Step 2.2.3

The integral of $\frac{1}{y}$ with respect to $y$ is $\ln (| y |)$ .

$e^{2 (\ln (| y |) + C)}$

Step 2.2.4

Simplify.

$e^{2 \ln (| y |) + C}$

Step 2.3

Remove the constant of integration.

$e^{2 \ln (y)}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (y^{2})}$

Step 2.5

Exponentiation and log are inverse functions.

$y^{2}$

Step 3

Multiply each term by the integrating factor $y^{2}$ .

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Step 3.1

Multiply each term by $y^{2}$ .

$y^{2} \frac{d x}{d y} + y^{2} (\frac{2 y}{y^{2}} x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2

Simplify each term.

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Step 3.2.1

Cancel the common factor of $y$ and $y^{2}$ .

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Step 3.2.1.1

Factor $y$ out of $2 y$ .

$y^{2} \frac{d x}{d y} + y^{2} (\frac{y \cdot 2}{y^{2}} x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $y$ out of $y^{2}$ .

$y^{2} \frac{d x}{d y} + y^{2} (\frac{y \cdot 2}{y \cdot y} x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.1.2.2

Cancel the common factor.

$y^{2} \frac{d x}{d y} + y^{2} (\frac{y \cdot 2}{y \cdot y} x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.1.2.3

Rewrite the expression.

$y^{2} \frac{d x}{d y} + y^{2} (\frac{2}{y} x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.2

Combine $\frac{2}{y}$ and $x$ .

$y^{2} \frac{d x}{d y} + y^{2} \frac{2 x}{y} = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.3

Cancel the common factor of $y$ .

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Step 3.2.3.1

Factor $y$ out of $y^{2}$ .

$y^{2} \frac{d x}{d y} + y \cdot y \frac{2 x}{y} = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.3.2

Cancel the common factor.

$y^{2} \frac{d x}{d y} + y \cdot y \frac{2 x}{y} = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.3.3

Rewrite the expression.

$y^{2} \frac{d x}{d y} + y (2 x) = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.2.4

Rewrite using the commutative property of multiplication.

$y^{2} \frac{d x}{d y} + 2 y x = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.3

Cancel the common factor of $y^{2}$ .

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Step 3.3.1

Cancel the common factor.

$y^{2} \frac{d x}{d y} + 2 y x = y^{2} \frac{\sin (y)}{y^{2}}$

Step 3.3.2

Rewrite the expression.

$y^{2} \frac{d x}{d y} + 2 y x = \sin (y)$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d y} [y^{2} x] = \sin (y)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d y} [y^{2} x] d y = \int \sin (y) d y$

Step 6

Integrate the left side.

$y^{2} x = \int \sin (y) d y$

Step 7

The integral of $\sin (y)$ with respect to $y$ is $- \cos (y)$ .

$y^{2} x = - \cos (y) + C$

Step 8

Divide each term in $y^{2} x = - \cos (y) + C$ by $y^{2}$ and simplify.

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Step 8.1

Divide each term in $y^{2} x = - \cos (y) + C$ by $y^{2}$ .

$\frac{y^{2} x}{y^{2}} = \frac{- \cos (y)}{y^{2}} + \frac{C}{y^{2}}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $y^{2}$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{y^{2} x}{y^{2}} = \frac{- \cos (y)}{y^{2}} + \frac{C}{y^{2}}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \cos (y)}{y^{2}} + \frac{C}{y^{2}}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$x = - \frac{\cos (y)}{y^{2}} + \frac{C}{y^{2}}$